#246327

order	512
nilpotency class	3
isoclinism	isoclinic class of #246319
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
prime radical	#46355	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1428 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210564, #72251753
maximal normal subgroup of	#72210564, #72251753

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153523);