#7694919

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

7240 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)(7,8)(9,12)(10,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,26)(24,27)(25,28);(1,2)(3,4)(13,15)(14,17)(16,22)(18,23)(19,20)(21,25)(24,26)(27,28);(6,11)(7,12)(14,16)(15,20)(21,24)(23,28);(5,7,10,12)(6,8,11,9)(13,16,19,14)(15,22,20,17)(18,24,27,21)(23,26,28,25);(5,8,10,9)(6,7,11,12)(13,17,19,22)(14,20,16,15)(18,26,27,25)(21,23,24,28);(5,9,10,8)(6,12,11,7)(13,18,19,27)(14,21,16,24)(15,23,20,28)(17,25,22,26)

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=h,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, 7602115);