#7648439

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

8352 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)(6,9)(7,11)(13,14)(15,20)(16,18)(17,24)(19,23)(21,25)(22,26)(27,28);(1,2)(3,4)(6,9)(10,12);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,20)(17,22)(18,23)(21,26)(24,27)(25,28);(5,7,8,11)(6,10,9,12)(13,16,18,14)(15,20,23,19)(17,24,25,21)(22,27,28,26);(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,26)(25,28);(13,17,18,25)(14,21,16,24)(15,22,23,28)(19,26,20,27)

Polycyclic presentation:

2^9|c^2=i,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555635);