#405458

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1498 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318058, #72318245, #72318301
maximal normal subgroup of	#72318058, #72318245, #72318301

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312654);