#346549

order	512
nilpotency class	3
isoclinism	isoclinic class of #346518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2182 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 2182 conjugacy classes of subgroups — too many to draw. Only its 224 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253745);