#356505

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gi,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263701);