#453976

order	512
nilpotency class	3
isoclinism	isoclinic class of #453974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2254 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)(7,12)(8,19)(9,14)(10,24)(11,16)(13,21)(15,26)(17,33)(18,31)(20,36)(22,32)(23,28)(25,30)(27,34)(29,35);(2,4)(5,7,11,9)(6,12,16,14)(8,17,22,20)(10,23,26,18)(13,29,30,27)(15,31,24,28)(19,33,32,36)(21,35,25,34);(1,2)(3,4)(5,8)(6,13)(7,17)(9,20)(10,25)(11,22)(12,27)(14,29)(15,32)(16,30)(18,35)(19,24)(21,26)(23,34)(28,36)(31,33);(2,4)(5,9,11,7)(6,14,16,12)(8,20,22,17)(10,18,26,23)(13,29,30,27)(15,28,24,31)(19,33,32,36)(21,34,25,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312497, #72312585
maximal normal subgroup of	#72312497, #72312585

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361172);