#635988

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

7482 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:

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

Polycyclic presentation:

2^9|c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304737, #72304741, #72304744
maximal normal subgroup of	#72304737, #72304741, #72304744
maximal quotient of	#72304977, #72305070

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 543184);