#233396

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,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, 140592);