#8075439

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

6993 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)(15,22)(16,20)(17,19)(18,21)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,22)(18,23)(19,20)(21,24)(25,27)(26,28);(5,6,9,10)(7,11,12,8)(14,17)(15,20)(18,26)(24,27);(7,12)(8,11)(13,16,19,22)(14,20,17,15)(18,27,26,24)(21,23,25,28);(5,7)(6,8)(9,12)(10,11)(13,17,19,14)(15,22,20,16)(18,25,26,21)(23,27,28,24);(5,8)(6,7)(9,11)(10,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,28)(22,27)

Polycyclic presentation:

2^9|a^2=gh,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, #72312837, #72315209
maximal normal subgroup of	#72312785, #72312837, #72315209

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982635);