#494878

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#481
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

792 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)(7,16,18,32)(8,13)(9,21)(10,25)(11,15)(12,19,28,34)(14,24)(17,33,26,31)(20,30)(22,35)(23,29,36,27);(1,2,4,3)(9,22)(14,30)(16,32)(19,34)(20,24)(21,35)(23,36)(31,33);(1,3,4,2)(5,7,10,26,11,18,8,17)(6,12,13,27,15,28,25,29)(9,23,24,19,22,36,20,34)(14,31,21,16,30,33,35,32);(5,8,11,10)(6,13,15,25)(7,17,18,26)(9,20,22,24)(12,27,28,29)(14,21,30,35)(16,33,32,31)(19,23,34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=eh,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,e^c=ei,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224792, #72226795, #72226797
maximal normal subgroup of	#72224792, #72226795, #72226797
maximal quotient of	C₄×#13138

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402074);