#7695794

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

6770 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7602990);