#270994

order	512
nilpotency class	3
isoclinism	isoclinic class of #270875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 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:

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

Polycyclic presentation:

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

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, 178190);