#7691834

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

6905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,e^2=hi,f^2=i,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, 7599030);