#430309

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1159 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)(5,6,11,16,10,15,12,9)(7,17,21,19,20,13,22,29)(8,23,25,31,27,32,26,14)(18,34,35,28,36,30,24,33);(5,7,11,21,10,20,12,22)(6,13,16,29,15,17,9,19)(8,24,25,18,27,35,26,36)(14,30,23,33,31,34,32,28);(5,8)(6,14)(7,18)(9,23)(10,27)(11,26)(12,25)(13,28)(15,31)(16,32)(17,33)(19,30)(20,36)(21,24)(22,35)(29,34);(2,4)(5,9,12,15,10,16,11,6)(7,19,22,17,20,29,21,13)(8,23,25,31,27,32,26,14)(18,30,35,33,36,34,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252700, #72256876, #72256881
maximal normal subgroup of	#72252700, #72256876, #72256881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337505);