#4949447

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3750 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333981, #72334001, #72334017
maximal normal subgroup of	#72333981, #72334001, #72334017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4856643);