#419832

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1659 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318090, #72318259, #72318301
maximal normal subgroup of	#72318090, #72318259, #72318301

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327028);