#188150

order	512
nilpotency class	3
isoclinism	isoclinic class of #188035
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2790 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 2790 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) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95346);