#7994845

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	#43919	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7902041);