#8069697

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

7278 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)(6,10)(7,11)(13,14,19,21)(15,20,25,18)(16,22,24,28)(17,23,27,26);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,19,25)(14,20,21,18)(16,17,24,27)(22,23,28,26);(6,10)(7,11)(14,21)(18,20)(22,28)(23,26);(7,11)(8,12)(13,16)(14,22)(15,17)(18,26)(19,24)(20,23)(21,28)(25,27);(5,7)(6,8)(9,11)(10,12)(13,17)(14,23)(15,24)(16,25)(18,22)(19,27)(20,28)(21,26);(5,8)(6,7)(9,12)(10,11)(13,18)(14,15)(16,26)(17,22)(19,20)(21,25)(23,24)(27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976893);