#7718095

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4592 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:

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

Polycyclic presentation:

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

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, 7625291);