#417741

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2312 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324937);