C₂×#7129 #124116

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#7129

Subgroups

	subgroup	G/H
center	C₂×C₄	#329
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#376	C₂³
socle	C₂²	#1272
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

488 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213057, #72214507, #72215066, …
maximal normal subgroup of	#72213057, #72214507, #72215066, …
maximal quotient of	C₂×#93377, #72279592, #72279620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31312);