#7692408

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

7355 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)(5,6,9,12)(7,10,8,11)(13,14)(15,23)(16,18)(17,22)(19,21)(20,24)(25,28)(26,27);(5,6)(7,10)(8,11)(9,12)(13,15,19,24)(14,20,21,23)(16,25,26,17)(18,22,27,28);(1,2,3,4)(5,7,9,8)(6,10,12,11)(14,21)(15,24)(16,26)(22,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,16)(14,18)(15,25)(17,24)(19,26)(20,22)(21,27)(23,28);(5,8,9,7)(6,11,12,10)(13,17,19,25)(14,22,21,28)(15,16,24,26)(18,20,27,23);(13,18)(14,16)(15,22)(17,23)(19,27)(20,25)(21,26)(24,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=gi,e^2=g,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, 7599604);