#248613

order	512
nilpotency class	3
isoclinism	isoclinic class of #248599
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	#39464	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155809);