#6459998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,8,9)(7,10,11,12)(13,14,18,25,20,26,19,24)(15,27,31,21,32,38,30,39)(16,22,33,40,34,41,17,23)(28,43,29,36,35,37,44,42);(6,9)(10,12)(13,15)(14,21)(16,29)(17,35)(18,30)(19,31)(20,32)(22,37)(23,42)(24,38)(25,27)(26,39)(28,33)(34,44)(36,40)(41,43);(1,2,3,4)(13,16,19,17,20,34,18,33)(14,22,24,23,26,41,25,40)(15,28,30,44,32,35,31,29)(21,36,27,43,39,42,38,37);(5,7,8,11)(6,10,9,12)(13,17,19,34,20,33,18,16)(14,23,24,41,26,40,25,22)(15,29,30,28,32,44,31,35)(21,37,27,36,39,43,38,42);(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,33,34,17)(21,38,39,27)(22,23,41,40)(28,44,35,29)(36,37,42,43)

Polycyclic presentation:

2^9|a^2=fi,c^2=ghi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367194);