C₂×#7427 #150260

order	512
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7427

Subgroups

	subgroup	G/H
center, socle	C₂²	#1351
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	#381	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁵ ◁ C₂×#381 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

515 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 74 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);(3,4,7,11,8,12,21,28)(5,13,15,29,16,30,32,34)(6,17,19,33,20,10,25,26)(9,22,23,14,24,31,27,18);(3,5)(4,9)(6,18,20,14)(7,15)(8,16)(10,13,17,30)(11,27)(12,24)(19,31,25,22)(21,32)(23,28)(26,29,33,34)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#150395, #72285100, #72285169, …
maximal normal subgroup of	C₂×#150395, #72285100, #72285169, …
maximal quotient of	#72285100

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57456);