#7185304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#30446	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092500);