#7124100

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61270	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7031296);