#7918533

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

8108 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)(5,6,9,11)(7,8,12,10)(14,19)(15,21)(16,24)(26,28);(1,2)(3,4)(13,14)(15,22)(16,20)(17,21)(18,19)(23,28)(24,27)(25,26);(5,7,9,12)(6,10,11,8)(13,15,18,21)(14,17,19,22)(16,25,24,23)(20,28,27,26);(5,7)(6,8)(9,12)(10,11)(13,16,18,24)(14,20,19,27)(15,23,21,25)(17,26,22,28);(5,8)(6,7)(9,10)(11,12)(13,14)(15,17)(16,20)(18,19)(21,22)(23,26)(24,27)(25,28);(13,17)(14,21)(15,19)(16,26)(18,22)(20,25)(23,27)(24,28)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^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	#72304329, C₂×#7918533
maximal normal subgroup of	#72304329, C₂×#7918533
maximal quotient of	C₂×#7918533

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825729);