#7693667

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

8195 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,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, 7600863);