#422606

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1071 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)(7,16,20,32)(8,13)(9,14)(10,26)(11,15)(12,19,28,21)(17,33,18,29)(22,35,34,27)(23,30)(24,36)(25,31);(5,7,9,21,11,20,10,19)(6,12,14,32,15,28,26,16)(8,22,23,18,25,34,24,17)(13,29,30,27,31,33,36,35);(5,8,9,23,11,25,10,24)(6,13,14,30,15,31,26,36)(7,17,19,34,20,18,21,22)(12,27,16,29,28,35,32,33);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,30,15,31,26,36)(7,18,19,22,20,17,21,34)(12,27,16,29,28,35,32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252923, #72252972, #72252977
maximal normal subgroup of	#72252923, #72252972, #72252977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329802);