#6401029

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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

1767 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)(15,25,26,20)(16,22,32,40)(17,24,19,21)(27,38,30,35)(28,36,29,34)(31,37,33,39)(41,43,42,44);(5,7,10,6)(8,12,9,11)(13,15)(14,20)(16,30)(17,28)(18,26)(19,29)(21,36)(22,38)(23,25)(24,34)(27,32)(31,42)(33,41)(35,40)(37,43)(39,44);(5,8,10,9)(6,11,7,12)(14,21)(15,26)(17,19)(20,34)(22,37)(23,24)(25,36)(27,30)(31,33)(35,43)(38,44)(39,40);(5,8,10,9)(6,12,7,11)(13,16,18,32)(14,22,23,40)(15,27,26,30)(17,33,19,31)(20,35,25,38)(21,37,24,39)(28,42,29,41)(34,43,36,44);(1,3,4,2)(5,9,10,8)(6,12,7,11)(13,17,18,19)(14,21,23,24)(15,28,26,29)(16,31,32,33)(20,36,25,34)(22,39,40,37)(27,41,30,42)(35,43,38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308225);