#303160

order	512
nilpotency class	3
isoclinism	isoclinic class of #267755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1086 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 160 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)(9,10,12,14)(11,16,15,13)(17,18,21,24)(19,25,27,32)(20,29,26,23)(22,30,31,28);(1,3,5,4)(2,6,8,7)(10,14)(13,16);(1,3,5,4)(2,7,8,6)(9,11,13,10,12,15,16,14)(17,19,26,30,21,27,20,28)(18,22,29,32,24,31,23,25);(1,4,5,3)(2,7,8,6)(10,14)(11,15)(19,27)(22,31)(25,32)(28,30)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=efh,d^2=ei,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,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, 210356);