#7082914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×D₄²	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1633 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25,23,26,21,19)(20,24,28,31,29,32,30,27);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,30)(28,32)(29,31);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,30)(22,28)(23,29)(25,31)(26,32);(3,8)(5,7)(11,16)(13,15)(17,18,21,19,23,26,22,25)(20,24,30,27,29,32,28,31);(10,14)(13,15)(17,21,23,22)(18,25,26,19)(20,28,29,30)(24,27,32,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=h,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308919, #72308949
maximal normal subgroup of	#72308919, #72308949

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6990110);