C₂²×#2276 #7625210

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂²×#2276

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator	C₂	C₂⁴×C₄²
Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

14122 conjugacy classes of subgroups; only the 12322 normal subgroups are shown.

normal subgroups only

This group has 14122 conjugacy classes of subgroups — too many to draw. Only its 12322 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) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2);(3,4);(5,6,8,7)(9,10,12,11)(14,16)(18,19);(9,10,12,11)(13,14,17,16)(15,19,20,18);(5,7,8,6)(9,11,12,10)(13,15,17,20)(14,18,16,19);(13,16,17,14)(15,19,20,18)

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532406);