C₂×#12861 #493871

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12861

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1958 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1958 conjugacy classes of subgroups — too many to draw. Only its 258 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)(5,8)(6,7)(9,10)(11,12,15,20,16,21,13,17)(14,19,22,26,23,18,24,25);(4,7)(8,10)(11,13,16,15)(12,17,21,20)(14,22,23,24)(18,25,19,26);(3,5,6,9)(4,8,7,10)(12,18)(14,23)(17,25)(19,21)(20,26)(22,24)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235757, #72235763, C₂×#133995, …
maximal normal subgroup of	#72235757, #72235763, C₂×#133995, …
maximal quotient of	C₂×#135490

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401067);