#238604

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
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

973 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:

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

Polycyclic presentation:

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

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, 145800);