#609021

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(5,6)(7,8)(9,10)(11,12)(14,17)(18,20);(1,2)(3,4)(9,11,12,10)(13,14,16,17)(15,18,19,20);(2,4)(10,11);(6,8)(10,11)(14,17)(18,20);(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

2^9|b^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263563, #72263585, #72263586, …
maximal normal subgroup of	#72263563, #72263585, #72263586, …
maximal quotient of	C₄×#15258, #72298425, #72336037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516217);