#354787

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3534 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239720, #72249991, #72257160, …
maximal normal subgroup of	#72239720, #72249991, #72257160, …
maximal quotient of	#72239742, #72265491, #72317350, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261983);