#269053

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	#24378	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)(13,14,18,23,19,24,17,22)(15,20,25,28,26,21,16,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,24,21)(16,18,25,17)(22,27,23,28);(2,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,21)(15,17)(18,26)(19,25)(20,24)(22,27)(23,28);(5,7)(6,9)(8,12)(10,11)(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,28,21,27)

Polycyclic presentation:

2^9|a^2=f,b^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, 176249);