C₂²×#2644 #7627774

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#2644

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

7770 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 7770 conjugacy classes of subgroups — too many to draw. Only its 3490 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);(6,9)(10,11)(13,14)(15,18)(16,19)(17,20)(21,22,25,27)(23,28,24,26);(5,6,8,9)(7,10,12,11)(13,15)(14,16)(17,19)(18,20)(21,23,25,24)(22,26,27,28);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,28)(25,27);(5,7,8,12)(6,10,9,11)(13,16,17,18)(14,19,20,15)(21,24,25,23)(22,26,27,28)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#55103, C₂²×#57653, C₂²×#42566, …
maximal normal subgroup of	C₂²×#55103, C₂²×#57653, C₂²×#42566, …
maximal quotient of	C₂³×#2644

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534970);