#521671

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328443, #72328446, #72328457
maximal normal subgroup of	#72328443, #72328446, #72328457
maximal quotient of	#72331622, #72331623

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428867);