#8063099

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

5208 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)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18,23,25)(19,27,21,24)(20,26,30,22)(28,29,31,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,28,30,31)(22,29,26,32);(3,7)(4,8)(9,11,13,12)(10,14,16,15)(18,25)(22,26)(24,27)(29,32);(2,6)(4,8)(17,20)(18,26)(19,28)(21,31)(22,25)(23,30)(24,32)(27,29);(1,4)(2,3)(5,8)(6,7)(9,11,13,12)(10,14,16,15)(17,21,23,19)(18,24,25,27)(20,28,30,31)(22,32,26,29);(1,2)(3,4)(5,6)(7,8)(9,12,13,11)(10,15,16,14)(17,22,23,26)(18,20,25,30)(19,29,21,32)(24,28,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318787, #72319702, #72319710
maximal normal subgroup of	#72318787, #72319702, #72319710

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7970295);