#95338

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1906 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1906 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=gi,f^2=gi,b^a=bd,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.

Relationships

maximal subgroup of	#72268046
maximal normal subgroup of	#72268046
maximal quotient of	#72268048, #72268049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2534);