#7228823

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(27,34)(28,36)(29,30)(31,32)(33,35)(37,39)(38,40);(2,6)(3,4)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,31)(26,32,29,34)(28,35,38,39)(33,36,37,40);(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)(26,33)(27,35)(29,37)(30,38)(31,39)(32,36)(34,40);(2,6)(7,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29,30,26)(27,34,31,32)(28,37,38,33)(35,40,39,36);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,29)(32,34)(33,37)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7136019);