#115637

order	512
nilpotency class	3
isoclinism	isoclinic class of #115631
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	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1380 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1380 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,g^b=gh,d^c=dh,g^c=gi

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, 22833);