#249921

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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1177 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)(3,11)(4,12)(5,10)(6,9)(7,14)(8,16)(13,15)(17,18,22,25)(19,23,27,21)(20,24,29,31)(26,28,32,30);(1,3)(2,4)(5,7)(6,13)(8,15)(9,14)(10,11)(12,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32);(1,4)(2,7)(3,8)(5,11)(6,14)(9,12)(10,13)(15,16)(17,20,22,29)(18,24,25,31)(19,26,27,32)(21,28,23,30);(1,4)(2,3)(5,15)(6,14)(7,8)(9,13)(10,12)(11,16)(17,21,22,23)(18,19,25,27)(20,28,29,30)(24,26,31,32)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,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, 157117);