#7883025

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790221);