#7595325

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#45023	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1633 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,20,27,30)(22,31,29,26)(24,32,28,25);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,25)(20,28)(22,29)(24,30)(27,32);(10,14)(11,16)(17,19,22,25,21,27,29,32)(18,24,26,20,23,28,31,30);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,29,24,21,30,22,28)(18,25,31,19,23,32,26,27);(2,5)(6,8)(10,14)(13,15)(18,23)(20,30)(24,28)(26,31)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502521);