#6373530

order	512
nilpotency class	3
isoclinism	isoclinic class of #60454
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=di,e^b=ei,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265571, #72265633
maximal normal subgroup of	#72265571, #72265633

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280726);