#7692407

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

8499 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)(5,6,8,10)(7,9,12,11)(13,14)(15,22)(16,17)(18,21)(19,20)(23,28)(24,26)(25,27);(5,6)(7,9)(8,10)(11,12)(13,15,19,21)(14,18,20,22)(16,23,25,26)(17,24,27,28);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,20)(15,21)(16,25)(24,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,16)(14,17)(15,23)(18,24)(19,25)(20,27)(21,26)(22,28);(13,17)(14,16)(15,24)(18,23)(19,27)(20,25)(21,28)(22,26);(13,18)(14,21)(15,20)(16,24)(17,26)(19,22)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gi,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, 7599603);