#233408

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,14)(7,16,18,19)(8,17,20,12)(9,15,11,13)(21,22,27,31)(23,32,24,35)(25,30,26,29)(28,33,36,34);(2,4)(5,7)(6,8)(9,17)(10,18)(11,12)(13,19)(14,20)(15,16)(21,23,27,24)(22,28,31,36)(25,33,26,34)(29,35,30,32);(5,8)(6,12)(7,13)(9,19)(10,20)(11,16)(14,17)(15,18)(21,24,27,23)(22,28,31,36)(25,34,26,33)(29,35,30,32);(5,9,10,11)(6,13,14,15)(7,17,18,12)(8,19,20,16)(21,25,27,26)(22,29,31,30)(23,33,24,34)(28,35,36,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140604);