C₂²×#1839 #6371060

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#1839

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂²×C₄	C₂⁵
socle, prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3474 conjugacy classes of subgroups; only the 1762 normal subgroups are shown.

normal subgroups only

This group has 3474 conjugacy classes of subgroups — too many to draw. Only its 1762 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) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278256);