#7708792

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	#36624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4956 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615988);