#7677957

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	#45243	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4601 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585153);