#7694699

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

7000 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^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, 7601895);