#249932

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	#61620	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1143 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)(3,12)(4,14)(5,8)(6,11)(7,10)(9,16)(13,15)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,3,7,13)(2,5,10,4)(6,14,16,8)(9,15,11,12)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,4,7,5)(2,8,10,14)(3,9,13,11)(6,15,16,12)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(1,5,7,4)(2,3,10,13)(6,15,16,12)(8,9,14,11)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,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, 157128);