#6371455

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,10,16,11,13,9,15)(7,17,21,12,20,27,19,25)(8,14,24,30,22,28,23,29)(18,31,34,26,32,36,33,35);(1,2,3,4)(5,7)(6,12)(8,18)(9,21)(10,19)(11,20)(13,25)(14,26)(15,27)(16,17)(22,32)(23,34)(24,33)(28,35)(29,36)(30,31);(6,13)(9,10)(12,25)(14,28)(19,21)(23,24)(26,35)(33,34);(5,8)(6,14)(7,18)(9,23)(10,24)(11,22)(12,26)(13,28)(15,29)(16,30)(17,31)(19,33)(20,32)(21,34)(25,35)(27,36);(8,22)(14,28)(18,32)(23,24)(26,35)(29,30)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,f^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286046, #72286244, #72287280, …
maximal normal subgroup of	#72286046, #72286244, #72287280, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278651);