#7918718

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5142 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,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.

Relationships

maximal subgroup of	#72265944, #72265966, #72265967
maximal normal subgroup of	#72265944, #72265966, #72265967
maximal quotient of	#72266543, #72266544

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825914);