#7696447

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

7204 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,26)(17,22)(18,23)(19,24)(21,27)(25,28);(1,2,3,4)(13,15)(14,20)(16,17)(18,21)(19,25)(22,26)(23,27)(24,28);(6,9)(10,12)(13,16,19,27)(14,21,24,26)(15,23,25,17)(18,20,22,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,17)(18,20)(19,27)(22,28)(23,25)(24,26);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,16)(18,24)(19,23)(20,26)(21,28)(25,27);(13,18,19,22)(14,23,24,17)(15,21,25,26)(16,20,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238187, #72238209, #72238212
maximal normal subgroup of	#72238187, #72238209, #72238212

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603643);