#7681746

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	#44239	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4755 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=gh,f^2=hi,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, 7588942);