#7604319

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=h,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7511515);