C₂×#31077 #7503370

order	512
nilpotency class	3
isoclinism	isoclinic class of #31077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#31077

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	#2679
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3488 conjugacy classes of subgroups; only the 586 normal subgroups are shown.

normal subgroups only

This group has 3488 conjugacy classes of subgroups — too many to draw. Only its 586 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 — 5 generators on 34 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#31077⋊C₃
maximal subgroup of	C₂×#125651, #72216206, #72216378, …
maximal normal subgroup of	C₂×#125651, #72216206, #72216378, …
maximal quotient of	C₂×#475763, #72213461, C₂²×#31077, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410566);