#93974

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	C₄⋊#95	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₄⋊#95 ◁ G
LECS	G ▷ C₄⋊#95 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 331 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=f,c^2=fi,d^2=g,e^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=def,e^b=ei,d^c=dhi,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, 1170);