#7185318

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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)(15,20,18,23)(16,27,30,34)(17,22,32,37)(21,25,36,28)(26,33,31,38)(29,40,42,43)(35,39,44,41);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15,19,18)(14,20,24,23)(16,28,30,25)(17,26,32,31)(21,34,36,27)(22,33,37,38)(29,41,42,39)(35,43,44,40);(5,7)(6,11)(8,10)(9,12)(13,16,15,25,19,30,18,28)(14,21,23,34,24,36,20,27)(17,29,26,39,32,42,31,41)(22,35,38,43,37,44,33,40);(2,4)(6,10)(7,12)(13,17)(14,22)(15,26)(16,29)(18,31)(19,32)(20,33)(21,35)(23,38)(24,37)(25,39)(27,40)(28,41)(30,42)(34,43)(36,44);(5,8,9,11)(6,7,10,12)(13,15,19,18)(14,23,24,20)(16,25,30,28)(17,31,32,26)(21,34,36,27)(22,33,37,38)(29,41,42,39)(35,40,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 7092514);