#6401003

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14,18,23)(15,19,24,32)(16,21,27,35)(17,20,29,22)(25,31,37,42)(26,33,38,30)(28,34,40,36)(39,43,44,41);(5,7,10,6)(8,12,9,11)(13,15)(14,19)(16,25)(17,26)(18,24)(20,33)(21,31)(22,30)(23,32)(27,37)(28,39)(29,38)(34,43)(35,42)(36,41)(40,44);(5,8,10,9)(6,11,7,12)(14,20)(15,24)(17,29)(19,30)(21,34)(22,23)(25,37)(28,40)(31,41)(32,33)(35,36)(42,43);(5,8,10,9)(6,12,7,11)(13,16,18,27)(14,21,23,35)(15,25,24,37)(17,28,29,40)(19,31,32,42)(20,34,22,36)(26,39,38,44)(30,41,33,43);(1,3,4,2)(5,9,10,8)(6,12,7,11)(16,27)(21,35)(25,37)(28,40)(31,42)(34,36)(39,44)(41,43)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308199);