#493071

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1228 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239482, #72239528, #72239548, …
maximal normal subgroup of	#72239482, #72239528, #72239548, …
maximal quotient of	#72319057, #72319995, C₂×#493071

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400267);