#249914

order	512
nilpotency class	3
isoclinism	isoclinic class of #249879
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	#61758	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157110);