#7673677

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

8186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256668, #72256695, #72256700
maximal normal subgroup of	#72256668, #72256695, #72256700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580873);