#428082

order	512
nilpotency class	3
isoclinism	isoclinic class of #428081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15377
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237065, #72237072, #72237084, …
maximal normal subgroup of	#72237065, #72237072, #72237084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335278);