C₂×#12391 #489236

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12391

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

737 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396432);