#95105

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,e^2=i,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=cgi,e^b=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2301);