#191224

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224831, #72224834, #72224835, …
maximal normal subgroup of	#72224831, #72224834, #72224835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98420);