#95366

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = 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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

464 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,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, 2562);