#7643399

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

8774 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)(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,24)(17,19)(18,20)(21,26)(23,27)(25,28);(1,2)(3,4)(5,6,8,10)(7,9,12,11)(13,15,18,17)(14,19,20,22)(16,23,26,25)(21,27,24,28);(5,6)(7,9)(8,10)(11,12)(14,20)(19,22)(21,24)(27,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,23)(17,25)(18,26)(19,27)(20,24)(22,28);(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28);(13,17,18,15)(14,22,20,19)(16,25,26,23)(21,28,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314806, #72315002
maximal normal subgroup of	#72314806, #72315002

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550595);