#7942751

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

8395 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)(5,6,9,11)(7,8,12,10)(13,14)(15,18)(16,21)(17,20)(19,24)(22,26)(23,28)(25,27);(1,2)(3,4)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28);(5,7,9,12)(6,10,11,8)(14,15)(19,21)(20,22)(27,28);(5,7)(6,8)(9,12)(10,11)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27);(5,8)(6,7)(9,10)(11,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256671, #72256675, #72256679, …
maximal normal subgroup of	#72256671, #72256675, #72256679, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7849947);