#6384018

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61030	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2531 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)(5,6,12,17,11,14,10,28)(7,18,22,13,19,32,23,31)(8,15,24,30,21,33,29,35)(9,25,20,36,26,16,27,34);(5,7)(6,13)(8,9)(10,23)(11,19)(12,22)(14,31)(15,34)(16,35)(17,32)(18,28)(20,24)(21,26)(25,30)(27,29)(33,36);(2,4)(6,14)(7,19)(9,26)(15,33)(17,28)(20,27)(22,23)(30,35);(5,8)(6,15)(7,20)(9,23)(10,29)(11,21)(12,24)(13,16)(14,33)(17,30)(18,36)(19,27)(22,26)(25,31)(28,35)(32,34);(5,9,12,20,11,26,10,27)(6,16,17,34,14,25,28,36)(7,21,22,29,19,8,23,24)(13,30,32,33,31,35,18,15)

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343392, #72343648, #72343653
maximal normal subgroup of	#72343392, #72343648, #72343653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291214);