#6384376

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2044 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)(6,9)(10,12)(13,14,17,22)(15,24,25,19)(16,21,29,37)(18,20,31,23)(26,36,28,33)(27,32,39,34)(30,35,42,38)(40,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,28)(17,25)(18,27)(20,34)(21,36)(22,24)(23,32)(26,29)(30,41)(31,39)(33,37)(35,43)(38,44)(40,42);(5,7,8,11)(6,10,9,12)(14,20)(15,25)(18,31)(19,32)(21,35)(22,23)(24,34)(26,28)(30,42)(33,43)(36,44)(37,38);(6,9)(10,12)(13,16,17,29)(14,21,22,37)(15,26,25,28)(18,30,31,42)(19,33,24,36)(20,35,23,38)(27,40,39,41)(32,43,34,44);(1,3,4,2)(16,29)(21,37)(26,28)(30,42)(33,36)(35,38)(40,41)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291572);