#115192

order	512
nilpotency class	3
isoclinism	isoclinic class of #115179
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 909 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) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22388);