#7988192

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	#45356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7895388);