#7643428

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
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550624);