C₂×#33324 #7622415

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₈ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#33324

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂	C₂³×C₄×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂³	C₂³×C₈
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 38 points:

(1,2);(7,8,10,15,11,16,20,28,12,14,21,26,22,27,32,36)(9,13,18,24,19,25,31,35,17,23,29,33,30,34,37,38);(3,4,5,6)(7,9)(8,13)(10,18)(11,19)(12,17)(14,23)(15,24)(16,25)(20,31)(21,29)(22,30)(26,33)(27,34)(28,35)(32,37)(36,38);(9,17)(13,23)(18,29)(19,30)(24,33)(25,34)(31,37)(35,38);(8,14)(13,23)(15,26)(16,27)(24,33)(25,34)(28,36)(35,38)

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#493129, C₂×#493066, C₂×#495141, …
maximal normal subgroup of	C₂×#493129, C₂×#493066, C₂×#495141, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529611);

C2×#33324 #7622415