#7693654

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

7009 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)(5,6,9,12)(7,10,8,11)(13,14)(15,25)(16,17)(18,22)(19,23)(20,24)(21,28)(26,27);(5,6)(7,10)(8,11)(9,12)(13,15)(14,21)(16,23)(17,27)(18,26)(19,22)(20,28)(24,25);(2,3)(13,16,20,18)(14,17,24,22)(15,26,28,23)(19,25,27,21);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,17,20,22)(14,16,24,18)(15,27,28,19)(21,23,25,26);(5,7,9,8)(6,10,12,11)(13,18,20,16)(14,22,24,17)(15,26,28,23)(19,25,27,21);(5,8,9,7)(6,11,12,10)(13,19,20,27)(14,23,24,26)(15,22,28,17)(16,25,18,21)

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=gh,e^2=g,f^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, 7600850);