C₂²×#2255 #7623211

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂²×#2255

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1458 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 1458 conjugacy classes of subgroups — too many to draw. Only its 738 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);(5,6,8,10)(7,11,12,9)(14,18)(15,20)(16,22)(19,24)(21,28)(23,27)(25,26);(5,7,8,12)(6,9,10,11)(14,19)(18,24)(21,25)(26,28);(13,14,20,26,16,21,23,24,17,19,27,28,22,25,15,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#33746, C₂²×#33912, C₂²×#33951
maximal normal subgroup of	C₂²×#33746, C₂²×#33912, C₂²×#33951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530407);