#7685128

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

7407 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)(6,10)(11,12)(13,14)(15,17)(16,22)(18,23)(19,21)(20,24)(25,26)(27,28);(5,6)(7,11)(8,12)(9,10)(13,15)(14,20)(16,18)(17,21)(19,24)(22,28)(23,26)(25,27);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,21)(15,24)(16,25)(23,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,22)(15,18)(17,23)(19,25)(20,28)(21,26)(24,27);(5,8,9,7)(6,12,10,11)(13,17,19,20)(14,15,21,24)(16,23,25,28)(18,26,27,22);(13,18)(14,23)(15,16)(17,22)(19,27)(20,26)(21,28)(24,25)

Polycyclic presentation:

2^9|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, 7592324);