#415475

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224916, #72224931, #72226816
maximal normal subgroup of	#72224916, #72224931, #72226816
maximal quotient of	#72224905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322671);