#188056

order	512
nilpotency class	3
isoclinism	isoclinic class of #188049
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#540	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2744 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 2744 conjugacy classes of subgroups — too many to draw. Only its 318 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95252);