#7708802

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	#36318	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)(10,14)(15,16)(17,18)(19,27)(20,24)(21,28)(22,29)(23,26)(25,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,26)(19,27,28,21)(20,24,30,32)(22,25,31,29);(2,5)(6,8)(17,19,23,28)(18,21,26,27)(20,29,30,25)(22,24,31,32);(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,25)(21,22)(23,30)(26,32)(27,31)(28,29);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,21)(18,19)(20,22)(23,27)(24,25)(26,28)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(17,22)(18,25)(19,24)(20,21)(23,31)(26,29)(27,30)(28,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,e^2=gh,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, 7615998);