#93647

order	512
nilpotency class	4
isoclinism	isoclinic class of #93523
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	C₂²×C₄×D₄	C₂²
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 505 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) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

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

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, 843);