C₂×#12301 #484572

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12301

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂×#3199	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 904 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);(3,4,7,23,8,13,24,12)(5,14,17,29,18,28,16,10)(6,19,21,11,22,9,20,26)(15,27,33,31,34,30,32,25);(4,9)(6,20)(7,24)(10,25)(11,23)(12,26)(13,19)(14,30)(15,32)(16,17)(21,22)(27,28)(29,31)(33,34);(3,5)(4,10)(6,15)(7,17)(8,18)(9,25)(11,30)(12,28)(13,29)(14,23)(16,24)(19,31)(20,32)(21,33)(22,34)(26,27)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486701, C₂×#485590, C₂×#486077, …
maximal normal subgroup of	C₂×#486701, C₂×#485590, C₂×#486077, …
maximal quotient of	#72265435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391768);