#269079

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176275);