#7574874

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,23)(22,31)(25,29)(27,32)(28,30);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,25,21,26)(20,30,31,27)(22,28,24,32);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,22)(19,27)(21,24)(23,31)(25,32)(26,28)(29,30);(10,13)(14,16)(17,21,23,18)(19,27,29,30)(20,22,31,24)(25,32,26,28);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,20,21,31)(19,28,29,32)(25,30,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7482070);