C₂²×#2070 #6432150

order	512
nilpotency class	3
isoclinism	isoclinic class of #2069
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#2070

Subgroups

	subgroup	G/H
center, socle	C₂³	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3034 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 3034 conjugacy classes of subgroups — too many to draw. Only its 914 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) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339346);