C₂²×#2381 #7625928

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂²×#2381

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator	C₂²	C₂⁵×C₄
Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8530 conjugacy classes of subgroups; only the 5634 normal subgroups are shown.

normal subgroups only

This group has 8530 conjugacy classes of subgroups — too many to draw. Only its 5634 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 32 points:

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

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, 7533124);