#778623

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3198 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)(5,6,9,8)(7,11,12,10)(13,14,17,18)(15,16,20,19)(21,22,25,27)(23,24,26,28);(5,7,9,12)(6,10,8,11)(14,18)(15,20)(21,22)(23,24)(25,27)(26,28);(5,6,9,8)(7,11,12,10)(13,15)(14,19)(16,18)(17,20)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,17,19)(14,15,18,20)(21,22)(23,24)(25,27)(26,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331996, #72332005, #72332056
maximal normal subgroup of	#72331996, #72332005, #72332056
maximal quotient of	#72333141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 685819);