C₂×#11453 #480037

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#11453

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1662 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1662 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387233);