#332977

order	512
nilpotency class	3
isoclinism	isoclinic class of #332911
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

677 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:

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

Polycyclic presentation:

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

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, 240173);