C₂×#11993 #483303

order	512
nilpotency class	4
isoclinism	isoclinic class of #11992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#11993

Subgroups

	subgroup	G/H
center, socle	C₂²	#1555
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1002 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 210 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);(3,4,6,12,7,11,20,27,8,10,19,26,22,13,30,29)(5,14,15,31,16,23,33,34,17,25,32,28,21,9,18,24);(3,5,8,17)(4,9,10,23)(6,18,19,33)(7,21,22,16)(11,25,13,14)(12,28,26,31)(15,20,32,30)(24,27,34,29);(4,10)(6,19)(9,23)(11,13)(14,25)(15,32)(18,33)(20,30)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217759
maximal normal subgroup of	#72217759
maximal quotient of	C₂×#128918, C₂×#124834, C₂×#124911

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390499);