#7229986

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fg,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,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, 7137182);