#7659055

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8560 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566251);