#7592164

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,30)(21,23)(22,31)(24,29)(26,27)(28,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,20)(22,27)(24,25)(28,29)(30,32);(1,3,4,7)(2,6,5,8)(17,19,27,20,21,28,22,29)(18,24,31,32,23,30,26,25);(11,15)(13,16)(17,20,27,28,21,29,22,19)(18,25,31,24,23,32,26,30);(2,5)(6,8)(18,23)(19,28)(20,29)(26,31)

Polycyclic presentation:

2^9|a^2=ghi,c^2=fg,d^2=gi,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7499360);