#7760094

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4598 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7667290);