#7510015

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7417211);