#7691822

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

6885 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)(8,12)(9,11)(13,14,18,17)(15,19,24,22)(16,20,26,25)(21,27,23,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,23)(17,22)(18,24)(20,28)(21,26)(25,27);(5,8,9,7)(6,10,11,12)(13,16)(14,20)(15,21)(17,25)(18,26)(19,27)(22,28)(23,24);(5,6)(7,10)(8,12)(9,11)(13,17,18,14)(15,22,24,19)(16,25,26,20)(21,28,23,27);(13,16)(14,20)(15,23)(17,25)(18,26)(19,28)(21,24)(22,27);(13,15)(14,19)(16,21)(17,22)(18,24)(20,27)(23,26)(25,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^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, 7599018);