#493387

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3418 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)(5,6,10,15,11,16,8,13)(7,17,18,34,21,32,19,35)(9,23,22,33,25,14,26,31)(12,27,29,24,30,36,28,20);(5,7)(6,12)(8,18)(9,24)(10,19)(11,21)(13,28)(14,32)(15,29)(16,30)(17,23)(20,25)(22,27)(26,36)(31,35)(33,34);(2,4)(5,8,11,10)(6,13,16,15)(7,18,21,19)(9,22,25,26)(12,28,30,29)(14,31,23,33)(17,34,32,35)(20,36,24,27);(5,8,11,10)(6,13,16,15)(7,19,21,18)(9,22,25,26)(12,29,30,28)(14,31,23,33)(17,35,32,34)(20,27,24,36)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260453, #72261190, #72261199, …
maximal normal subgroup of	#72260453, #72261190, #72261199, …
maximal quotient of	#72238533

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400583);