#6370734

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
prime radical	C₂²×#585	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

17294 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341483, #72341494, #72341508, …
maximal subgroup of	#72238117, #72249705, #72249722, …
maximal normal subgroup of	#72238117, #72249705, #72249722, …
maximal quotient of	#72238122, #72251215, #72306049, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277930);