#7693677

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

8415 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)(13,14,19,23)(15,24,27,20)(16,21,28,18)(17,22,25,26);(5,6,9,10)(7,8,11,12)(13,15,19,27)(14,20,23,24)(16,25,28,17)(18,26,21,22);(2,3)(5,7)(6,8)(9,11)(10,12)(13,16,19,28)(14,18,23,21)(15,17,27,25)(20,22,24,26);(1,3,4,2)(5,8)(6,11)(7,10)(9,12)(13,16)(14,21)(15,25)(17,27)(18,23)(19,28)(20,22)(24,26);(13,17,19,25)(14,22,23,26)(15,16,27,28)(18,24,21,20);(13,18,19,21)(14,16,23,28)(15,26,27,22)(17,20,25,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,e^2=i,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, 7600873);