#7064365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(7,9)(8,10)(11,12)(13,14,18,23,17,22,30,20)(15,24,27,34,25,19,31,32)(16,21,29,36,28,37,41,35)(26,38,40,44,39,33,42,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,17,25)(14,19,22,24)(16,26,28,39)(18,27,30,31)(20,34,23,32)(21,33,37,38)(29,40,41,42)(35,44,36,43);(5,7,8,12)(6,9,10,11)(14,20)(15,25)(16,28)(18,30)(19,32)(21,36)(22,23)(24,34)(33,44)(35,37)(38,43)(40,42);(1,3,4,2)(7,12)(9,11)(13,16)(14,21)(15,26)(17,28)(18,29)(19,33)(20,35)(22,37)(23,36)(24,38)(25,39)(27,40)(30,41)(31,42)(32,43)(34,44);(6,10)(9,11)(14,22)(15,25)(16,28)(20,23)(27,31)(29,41)(33,38)(43,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=hi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971561);