#7650329

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

7284 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)(13,14)(15,20)(16,18)(17,23)(19,21)(22,24)(25,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,20,21,26)(16,17,24,27)(18,23,22,28);(6,9)(10,12)(14,21)(18,22)(20,26)(23,28);(7,11)(10,12)(13,16)(14,22)(15,17)(18,21)(19,24)(20,28)(23,26)(25,27);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,24)(16,25)(18,26)(19,27)(20,22)(21,28);(5,6)(7,10)(8,9)(11,12)(13,18,19,22)(14,16,21,24)(15,23,25,28)(17,26,27,20)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238144, #72238189, #72238225
maximal normal subgroup of	#72238144, #72238189, #72238225
maximal quotient of	#72263479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7557525);