#420384

order	512
nilpotency class	3
isoclinism	isoclinic class of #420383
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15277
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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)(3,13,15,23)(4,9)(5,19,21,8)(6,10)(7,12)(11,20)(14,24,29,28)(16,22,31,32)(17,25)(18,27)(26,30);(1,3)(2,8)(4,16)(5,20)(6,21)(7,15)(9,24)(10,23)(11,13)(12,19)(14,17)(18,31)(22,26)(25,32)(27,28)(29,30);(1,4,6,17,7,18,20,30)(2,9,10,25,12,27,11,26)(3,14,5,16,15,29,21,31)(8,22,13,28,19,32,23,24);(1,5,20,3,7,21,6,15)(2,8,11,23,12,19,10,13)(4,16,30,14,18,31,17,29)(9,22,26,24,27,32,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338546, #72338550, #72338561
maximal normal subgroup of	#72338546, #72338550, #72338561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327580);