#269125

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#24384	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1363 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)(7,11,12,9)(8,10)(13,14,19,27,18,23,17,22)(15,20,16,26,25,21,24,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,25)(14,20,23,21)(16,17,24,19)(22,28,27,26);(2,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,24)(14,21,23,20)(15,17,25,19)(22,28,27,26);(5,7)(6,9)(8,12)(10,11)(13,17,18,19)(14,22,23,27)(15,16,25,24)(20,26,21,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=i,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, 176321);