C₂²×#2032 #6431844

order	512
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#2032

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4934 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 4934 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) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2);(3,4);(5,6)(7,10)(8,15,17,11)(9,12)(13,20,16,18)(14,19);(5,7)(6,10)(8,16,17,13)(9,14)(11,18,15,20)(12,19);(5,8)(6,11)(7,13)(9,17)(10,18)(12,15)(14,16)(19,20)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72258558, C₂²×#2032⋊C₃
maximal subgroup of	#72240509, #72240510, C₂×#432839, …
maximal normal subgroup of	#72240509, #72240510, C₂×#432839, …
maximal quotient of	#72266065, #72266109, C₂²×#10390, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339040);