#95015

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(17,18)(19,25,27,31)(20,23)(21,24)(22,26,29,32)(28,30)(33,34,36,39,37,40,43,46)(35,38,42,45,41,44,47,48);(1,3,5,13,6,14,4,12)(2,7,8,15,10,16,9,11)(17,19)(18,22,24,29)(20,26)(21,27)(23,25,30,31)(28,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(35,41)(38,44)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=ci,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, 2211);