C₂²×#2818 #7633398

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂²×#2818

Subgroups

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

14190 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 14190 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) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂²×#2818⋊C₃
maximal subgroup of	C₂×#5021378, C₂×#5020974, C₂²×#2818⋊C₃, …
maximal normal subgroup of	C₂×#5021378, C₂×#5020974, C₂²×#2818⋊C₃, …
maximal quotient of	C₂²×#16502, C₂²×#16662, C₂²×#17086, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540594);