#7618072

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518936
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	#31056	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(10,14)(11,12)(17,18,20,22)(19,25,28,23)(21,24,27,29)(26,30,32,31);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(18,22)(19,26)(21,27)(23,30)(25,31)(28,32);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16);(9,11,13,12)(10,15,14,16)(17,19,27,32,20,28,21,26)(18,23,29,30,22,25,24,31);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(18,22)(23,25)(24,29)(30,31)

Polycyclic presentation:

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

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, 7525268);