#95367

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
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	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

464 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)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,20,24,21,23,28,31)(19,25,27,32,26,22,29,30)(33,34)(35,43,45,48)(36,41)(37,42)(38,40)(39,44,47,46);(1,3)(2,7,10,15)(4,14)(5,12)(6,13)(8,16,9,11)(17,19)(18,22,23,25)(20,27)(21,26)(24,30,31,32)(28,29)(33,35,37,44,38,45,36,46)(34,39,42,43,40,47,41,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(18,23)(22,25)(24,31)(30,32)(34,40)(39,47)(41,42)(43,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=hi,f^2=i,b^a=bd,c^a=cgh,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2563);