#245696

order	512
nilpotency class	3
isoclinism	isoclinic class of #245615
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1417 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)(3,11)(4,12)(5,9)(6,22)(7,14)(8,15)(10,17)(13,16)(18,25)(19,30)(20,28)(21,26)(23,24)(27,32)(29,31);(2,9)(3,5)(10,12)(11,14)(13,25)(15,26)(17,20)(18,21)(22,28)(24,27)(29,30)(31,32);(1,3,16,21)(2,10,13,27)(4,17,23,32)(5,8,18,7)(6,20,19,31)(9,24,25,12)(11,30,26,22)(14,28,15,29);(1,4,16,23)(2,10,13,27)(3,17,21,32)(5,20,18,31)(6,8,19,7)(9,12,25,24)(11,22,26,30)(14,28,15,29)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213216
maximal normal subgroup of	#72213216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152892);