#4430405

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2613	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3997 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234636, #72234638, #72234646, …
maximal normal subgroup of	#72234636, #72234638, #72234646, …
maximal quotient of	#72237740, #72284248, #72328401

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4337601);