#7684492

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

7121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=hi,f^2=i,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, 7591688);