#298574

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1094 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205770);