#93645

order	512
nilpotency class	4
isoclinism	isoclinic class of #93514
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂³×C₂²⋊C₄	C₂²
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

778 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fhi,e^2=fh,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72236026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 841);