#6398615

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	C₂×#548	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1990 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,25,21)(16,27,30,39)(17,31,33,23)(20,32,37,29)(24,40,28,44)(26,38,42,35)(34,43,36,41);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,28)(17,26)(18,25)(20,36)(21,22)(23,35)(24,30)(27,44)(29,43)(31,38)(32,41)(33,42)(34,37)(39,40);(13,16,18,30)(14,20,22,37)(15,24,25,28)(17,32,33,29)(19,34,21,36)(23,39,31,27)(26,43,42,41)(35,44,38,40);(7,12)(9,11)(13,15)(14,21)(16,24)(17,26)(18,25)(19,22)(20,36)(23,38)(27,44)(28,30)(29,41)(31,35)(32,43)(33,42)(34,37)(39,40);(1,3,4,2)(6,10)(9,11)(14,22)(15,25)(20,37)(23,31)(24,28)(26,42)(27,39)(41,43)

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ci,d^b=df,e^b=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, 6305811);