#415503

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
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	#2487
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319978, #72320063, #72320074, …
maximal normal subgroup of	#72319978, #72320063, #72320074, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322699);