C₂×#12395 #489077

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12395

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3341	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

752 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 162 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);(3,4,8,12)(5,13,16,17)(6,11,7,10)(9,15,18,14)(20,25)(21,24)(22,29)(26,34)(27,33)(28,31)(30,32);(3,5,8,16)(4,9,12,18)(6,15,7,14)(10,17,11,13)(19,20,23,27)(21,26,24,32)(22,30,31,34)(25,28,33,29);(4,10)(6,7)(9,17)(11,12)(13,18)(14,15)(19,21,23,24)(20,26,27,32)(22,28,31,29)(25,30,33,34)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396273);