#400223

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321897
maximal normal subgroup of	#72321897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307419);