#2134893

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4324 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)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,19,20,17)(21,22)(23,28)(24,26)(25,27);(6,9)(10,12)(13,15)(14,17)(16,20)(18,19)(21,23,26,25)(22,27,24,28);(7,11)(10,12)(14,18)(17,19)(23,25)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,20)(18,19)(21,24,26,22)(23,27,25,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,23)(22,27,24,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=h,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	#72300528, #72300549, #72300562, …
maximal normal subgroup of	#72300528, #72300549, #72300562, …
maximal quotient of	#72330643, #72331318

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2042089);