#7185320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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)(15,25,26,35)(16,21,31,37)(17,23,18,22)(20,28,33,29)(27,39,40,44)(30,38,32,36)(34,42,43,41);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(15,26)(20,33)(25,35)(27,40)(28,29)(34,43)(39,44)(41,42);(5,7)(6,11)(8,10)(9,12)(13,15,17,28,19,26,18,29)(14,20,22,35,24,33,23,25)(16,27,32,42,31,40,30,41)(21,34,38,44,37,43,36,39);(2,4)(6,10)(7,12)(13,16)(14,21)(15,27)(17,32)(18,30)(19,31)(20,34)(22,38)(23,36)(24,37)(25,39)(26,40)(28,42)(29,41)(33,43)(35,44);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,22,24,23)(15,28,26,29)(16,30,31,32)(20,35,33,25)(21,36,37,38)(27,41,40,42)(34,39,43,44)

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=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092516);