#213283

order	512
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#19616	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 238 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210567
maximal normal subgroup of	#72210567
maximal quotient of	#72210396, #72210416, #72210421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120479);