#7645932

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

9050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|e^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267574, #72267581
maximal normal subgroup of	#72267574, #72267581

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553128);