C₄×#1710 #212885

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₄×#1710

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4);(5,6,9,11)(7,13,15,20)(8,16,17,12)(10,18,19,14)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36);(5,7,8,14,9,15,17,18)(6,10,12,13,11,19,16,20)(22,26)(27,34)(28,33)(29,35);(6,11)(10,19)(12,16)(13,20)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120081);