#299049

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fhi,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206245);