C₂²×#2030 #6431831

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#2030

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5054 conjugacy classes of subgroups; only the 954 normal subgroups are shown.

normal subgroups only

This group has 5054 conjugacy classes of subgroups — too many to draw. Only its 954 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#23987, C₂²×#23986, #72285100, …
maximal normal subgroup of	C₂²×#23987, C₂²×#23986, #72285100, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339027);