#450833

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

950 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252695, #72256869, #72256881
maximal normal subgroup of	#72252695, #72256869, #72256881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358029);