#239472

order	512
nilpotency class	3
isoclinism	isoclinic class of #232373
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	#37439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=di,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146668);