#7534262

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,19)(20,23,28,25)(22,24,29,26)(27,32,30,31);(2,6)(7,8)(11,16)(13,15)(18,19)(20,27)(22,29)(23,31)(25,32)(28,30);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,21,18)(20,25,28,23)(22,26,29,24)(27,31,30,32);(10,14)(13,15)(17,20,21,28)(18,23,19,25)(22,30,29,27)(24,31,26,32);(1,4,5,3)(2,7,6,8)(18,19)(23,25)(24,26)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7441458);