C₂²×#2785 #7633513

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄³

Aliases

direct product

C₂²×#2785

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8090 conjugacy classes of subgroups; only the 3314 normal subgroups are shown.

normal subgroups only

This group has 8090 conjugacy classes of subgroups — too many to draw. Only its 3314 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) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#42091, C₂²×#42517, C₂²×#52228, …
maximal normal subgroup of	C₂²×#42091, C₂²×#42517, C₂²×#52228, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540709);