#8075415

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

7086 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)(5,6,9,10)(7,11,12,8)(13,14,18,19)(15,20,21,16)(17,22,25,27)(23,28,26,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,28)(24,26)(25,27);(5,6,9,10)(7,11,12,8)(14,19)(16,20)(17,25)(23,26);(7,12)(8,11)(13,15,18,21)(14,20,19,16)(17,26,25,23)(22,24,27,28);(5,7)(6,8)(9,12)(10,11)(13,16,18,20)(14,21,19,15)(17,24,25,28)(22,26,27,23);(5,8)(6,7)(9,11)(10,12)(13,17)(14,22)(15,23)(16,24)(18,25)(19,27)(20,28)(21,26)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312785, #72312841, #72315206
maximal normal subgroup of	#72312785, #72312841, #72315206

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982611);