#7696495

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

8592 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,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, 7603691);