#493334

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3302 conjugacy classes of subgroups; only the 506 normal subgroups are shown.

normal subgroups only

This group has 3302 conjugacy classes of subgroups — too many to draw. Only its 506 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(5,6)(7,8)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(2,4)(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(6,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287145, #72287164, #72287172, …
maximal normal subgroup of	#72287145, #72287164, #72287172, …
maximal quotient of	#72287763

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400530);