#8934375

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	#60045	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5248 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72340591
maximal subgroup of	#72340574, #72340591
maximal normal subgroup of	#72340574, #72340591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841571);