#7187447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 7094643);