#7599984

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=ghi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^c=ei,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, 7507180);