#6370902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6370897
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#381
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4325 conjugacy classes of subgroups; only the 1296 normal subgroups are shown.

normal subgroups only

This group has 4325 conjugacy classes of subgroups — too many to draw. Only its 1296 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^b=hi,g^c=gi,e^d=ei

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278098);