#7697299

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

7255 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)(11,12)(13,14,19,18)(15,16,23,20)(17,21,25,26)(22,24,28,27);(5,6,9,10)(7,11,8,12)(13,15,19,23)(14,20,18,16)(17,22,25,28)(21,27,26,24);(2,3)(13,16,19,20)(14,15,18,23)(17,24,25,27)(21,22,26,28);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,16)(14,15)(17,24)(18,23)(19,20)(21,22)(25,27)(26,28);(5,8,9,7)(6,12,10,11)(13,17)(14,21)(15,22)(16,24)(18,26)(19,25)(20,27)(23,28);(13,18,19,14)(15,16,23,20)(17,21,25,26)(22,27,28,24)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=gi,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, 7604495);