C₂×#8545 #188031

order	512
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#8545

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2974 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 2974 conjugacy classes of subgroups — too many to draw. Only its 338 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 — 4 generators on 34 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235753, #72235802, C₂×#135486, …
maximal normal subgroup of	#72235753, #72235802, C₂×#135486, …
maximal quotient of	C₂×#133995

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95227);