C₂×#12277 #484854

order	512
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12277

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 795 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 50
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4,8,11)(5,12,14,17)(6,10,7,15)(9,16,18,13)(19,20)(21,25)(22,32)(23,36)(24,28)(26,35)(27,37)(29,44)(30,43)(31,40)(33,48)(34,47)(38,45)(39,46)(41,49)(42,50);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(19,21,24,31)(20,25,28,40)(22,33,34,29)(23,30,37,46)(26,41,42,38)(27,43,36,39)(32,44,47,48)(35,49,50,45);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(19,22)(20,26,28,42)(21,29)(23,35)(24,34)(25,38,40,41)(27,32,36,47)(30,45)(31,33)(37,50)(39,44,43,48)(46,49)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392050);