#7511112

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7418308);