#422921

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314443, #72314454, #72314468
maximal normal subgroup of	#72314443, #72314454, #72314468
maximal quotient of	#72314510

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330117);