C₂×#7402 #150180

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7402

Subgroups

	subgroup	G/H
center, socle	C₂³	#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂×#2011	C₂
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#390 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

906 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 26 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#150564, #72340618, #72340620
maximal normal subgroup of	C₂×#150564, #72340618, #72340620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57376);