C₂×#10525 #238930

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#10525

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

970 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 186 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)(5,9)(6,8)(7,11)(10,14)(12,15)(13,16)(17,18)(20,25)(21,28)(22,23)(26,29)(27,32)(30,31);(4,8)(5,12)(6,14)(9,16)(10,11)(13,18)(19,20)(21,29)(22,32)(23,25)(24,27)(26,31)(28,34)(30,33);(3,5,6,12,7,13,14,18)(4,9,10,17,11,16,8,15)(19,21,23,28,24,31,22,30)(20,26,32,34,27,29,25,33)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146126);