#7856369

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

8294 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)(13,14)(15,20)(16,17)(18,21)(19,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,6,9,12)(7,8,11,10)(13,15)(14,20)(16,27)(17,25)(18,24)(19,26)(21,28)(22,23);(5,7)(6,10)(8,12)(9,11)(13,16,19,18)(14,21,23,17)(15,24,26,27)(20,25,22,28);(13,17,19,21)(14,16,23,18)(15,25,26,28)(20,27,22,24);(5,7)(6,8)(9,11)(10,12)(13,15)(14,22)(16,24)(17,25)(18,27)(19,26)(20,23)(21,28);(5,8)(6,11)(7,12)(9,10)(13,18,19,16)(14,21,23,17)(15,24,26,27)(20,28,22,25)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256672, #72256679, #72256682
maximal normal subgroup of	#72256672, #72256679, #72256682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7763565);