#669913

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3462 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332355, #72332357, #72332388
maximal normal subgroup of	#72332355, #72332357, #72332388
maximal quotient of	#72333172

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 577109);