C₂×#33326 #7622417

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×#33326

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂	C₂³×C₄×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂³	C₂³×C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#512548, C₂×#479908, C₂×#483326, …
maximal normal subgroup of	C₂×#512548, C₂×#479908, C₂×#483326, …
maximal quotient of	#72231856, C₂×#479943, #72231892, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529613);

C2×#33326 #7622417