#494886

order	512
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#442
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

796 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264797, #72264833, #72264874, …
maximal normal subgroup of	#72264797, #72264833, #72264874, …
maximal quotient of	#72310166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402082);