#7626202

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂³×#120	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

17282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 6 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72236339, #72236358, #72240885, …
maximal subgroup of	#72217511, #72217818, #72218104, …
maximal normal subgroup of	#72217511, #72217818, #72218104, …
maximal quotient of	#72232292, #72232317, #72232450, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533398);