#2084233

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2668	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3262 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)(7,8)(9,12)(10,11)(13,14,16,18)(15,19,20,17)(21,22,24,26)(23,27,28,25);(5,7)(6,9)(8,11)(10,12)(15,20)(17,19)(21,23)(22,25)(24,28)(26,27);(5,8,10,9)(6,7,11,12)(13,14)(15,17)(16,18)(19,20)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,9,10,8)(6,7,11,12)(13,15)(14,17)(16,20)(18,19)(21,22)(23,25)(24,26)(27,28);(5,7)(6,8)(9,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256745, #72256757, #72256773, …
maximal normal subgroup of	#72256745, #72256757, #72256773, …
maximal quotient of	#72333003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1991429);