#2133642

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
prime radical	#16291	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=f,d^2=fgh,e^2=fgh,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	#72281123, #72281476, #72281940
maximal normal subgroup of	#72281123, #72281476, #72281940

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2040838);