#7906010

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	#37660	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7813206);