#358919

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266115);