#7694214

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

7247 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)(5,6,10,11)(7,8,12,9)(13,14)(15,16)(17,21)(18,20)(19,22)(23,24)(25,28)(26,27);(1,2)(3,4)(13,15)(14,19)(16,20)(17,23)(18,22)(21,25)(24,27)(26,28);(6,11)(7,12)(14,20)(15,22)(21,27)(23,28);(5,7)(6,8)(9,11)(10,12)(13,15)(14,19)(16,20)(17,23)(18,22)(21,25)(24,27)(26,28);(5,8,10,9)(6,12,11,7)(13,16,18,19)(14,15,20,22)(17,25,26,24)(21,28,27,23);(5,9,10,8)(6,7,11,12)(13,17,18,26)(14,21,20,27)(15,23,22,28)(16,24,19,25)

Polycyclic presentation:

2^9|a^2=gh,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, 7601410);