C₂×#12313 #488868

order	512
nilpotency class	4
isoclinism	isoclinic class of #12310
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12313

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

958 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,9,5)(6,16,19,32)(7,21,26,22)(8,15,24,12)(10,27,13,18)(11,30,14,25)(17,31,20,33)(23,28,34,29);(3,5,9,4)(6,13,19,10)(7,22,26,21)(8,12,24,15)(11,25,14,30)(16,23,32,34)(17,28,20,29)(18,33,27,31);(3,6)(4,10)(5,13)(7,23)(8,17)(9,19)(11,31)(12,28)(14,33)(15,29)(16,21)(18,25)(20,24)(22,32)(26,34)(27,30)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#488869, #72226757
maximal normal subgroup of	C₂×#488869, #72226757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396064);