#396484

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238736, #72264965, #72264972
maximal normal subgroup of	#72238736, #72264965, #72264972
maximal quotient of	#72238751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303680);