C₂×#14661 #544367

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄²

Aliases

direct product

C₂×#14661

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

9090 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.

normal subgroups only

This group has 9090 conjugacy classes of subgroups — too many to draw. Only its 1346 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) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#14661⋊C₃
maximal subgroup of	C₂×#5021378, C₂×#5020974, C₂×#5021372, …
maximal normal subgroup of	C₂×#5021378, C₂×#5020974, C₂×#5021372, …
maximal quotient of	C₂×#156474, C₂²×#14661, C₂×#155653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451563);