#213516

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1246 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 290 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120712);