#7307925

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,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, 7215121);