#95252

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×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

578 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2448);