#7698519

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

8203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7605715);