#7653986

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

8910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304561, #72304580, #72304601
maximal normal subgroup of	#72304561, #72304580, #72304601

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561182);