#184909

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#3459	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2575 conjugacy classes of subgroups; only the 356 normal subgroups are shown.

normal subgroups only

This group has 2575 conjugacy classes of subgroups — too many to draw. Only its 356 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)(10,13,16,15)(12,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(19,25)(22,28)(26,31)(29,32);(2,8)(6,9)(7,14)(13,15)

Polycyclic presentation:

2^9|a^2=g,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,c^b=ci,f^b=fh,g^b=gh,e^c=eh

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, 92105);