#7689761

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	#34655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5379 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)(3,7)(4,8)(5,6)(10,14)(15,16)(17,18,22,25)(19,23,27,20)(21,24,30,32)(26,29,31,28);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,25)(21,26)(22,27)(24,29)(28,32)(30,31);(2,6)(7,8)(17,19,22,27)(18,20,25,23)(21,26,30,31)(24,28,32,29);(1,2)(3,7)(4,8)(5,6)(17,19)(18,23)(20,25)(21,26)(22,27)(24,29)(28,32)(30,31);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,19,25,27)(21,29,30,28)(24,31,32,26);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,24,25,32)(19,26,27,31)(20,28,23,29)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596957);