#8077417

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	#45093	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4728 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984613);