C₂×#12299 #484567

order	512
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12299

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 854 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 — 4 generators on 34 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486689, C₂×#486080, C₂×#485583
maximal normal subgroup of	C₂×#486689, C₂×#486080, C₂×#485583
maximal quotient of	#72307570

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391763);