#7646409

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

8678 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)(6,9)(7,11)(13,14)(15,22)(16,25)(17,20)(18,26)(19,21)(23,24)(27,28);(1,2,3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(13,15,19,17)(14,20,21,22)(16,23,26,27)(18,24,25,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,23)(17,27)(19,26)(20,24)(21,25)(22,28);(13,17,19,15)(14,20,21,22)(16,23,26,27)(18,28,25,24);(13,18)(14,16)(15,24)(17,28)(19,25)(20,23)(21,26)(22,27)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=hi,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	#72256617, #72256682, #72256685
maximal normal subgroup of	#72256617, #72256682, #72256685
maximal quotient of	#72263615, #72263618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553605);