#7689105

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	#43278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4662 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)(9,10,14,12)(11,15,13,16)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,29,32,28);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,12,16)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,29,32,28);(1,2,5,4)(3,6,8,7)(10,12)(15,16)(18,23)(19,26)(25,31)(27,32);(9,12,14,10)(11,16,13,15)(17,19)(18,24)(20,23)(21,27)(22,26)(25,29)(28,31)(30,32);(9,13,14,11)(10,16,12,15)(17,20,22,24)(18,19,23,26)(21,29,30,28)(25,32,31,27);(1,4,5,2)(3,7,8,6)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596301);