#432896

order	512
nilpotency class	3
isoclinism	isoclinic class of #432895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15468
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1029 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338618, #72338620, #72338630
maximal normal subgroup of	#72338618, #72338620, #72338630

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340092);