#7967843

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	#40484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5147 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318056, #72319769, #72319886
maximal normal subgroup of	#72318056, #72319769, #72319886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875039);