#7689294

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

7186 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,e^2=h,f^2=h,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, 7596490);