#495710

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33516	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,11,16,12,18,34,17)(7,19,22,37,24,49,23,56)(8,14,27,42,28,44,59,43)(9,15,30,45,31,46,10,32)(13,35,38,55,39,33,47,21)(20,48,52,65,54,68,53,62)(25,40,26,57,29,41,60,66)(36,61,63,50,64,58,67,51);(5,7)(6,13)(8,20)(9,21)(10,33)(11,22)(12,24)(14,36)(15,37)(16,38)(17,47)(18,39)(19,32)(23,34)(25,50)(26,58)(27,52)(28,54)(29,51)(30,35)(31,55)(40,62)(41,65)(42,63)(43,67)(44,64)(45,49)(46,56)(48,57)(53,59)(60,61)(66,68);(2,4)(5,8)(6,14)(7,20)(9,29)(10,26)(11,27)(12,28)(13,36)(15,41)(16,42)(17,43)(18,44)(19,48)(21,51)(22,52)(23,53)(24,54)(25,31)(30,60)(32,57)(33,58)(34,59)(35,61)(37,65)(38,63)(39,64)(40,46)(45,66)(47,67)(49,68)(50,55)(56,62);(5,9,11,30,12,31,34,10)(6,15,16,45,18,46,17,32)(7,21,22,35,24,55,23,33)(8,25,27,26,28,29,59,60)(13,37,38,49,39,56,47,19)(14,40,42,57,44,41,43,66)(20,50,52,58,54,51,53,61)(36,62,63,48,64,65,67,68)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323138
maximal normal subgroup of	#72323138
maximal quotient of	#72323077, #72323125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402906);