#7830812

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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5508 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)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32);(1,3)(2,5)(4,8)(6,7)(17,19,22,25)(18,23,21,27)(20,26,28,31)(24,30,29,32);(2,6)(5,7)(9,10,12,14)(11,13,16,15)(19,25)(23,27)(26,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32);(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7738008);