#354876

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1394 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)(7,19,22,13)(8,24,26,31)(9,15)(10,17)(11,18)(12,16)(14,25,34,27)(20,35)(21,33,23,32)(28,36)(29,30);(5,7,11,23,12,22,10,21)(6,13,18,33,16,19,17,32)(8,20,27,9,26,28,25,29)(14,30,31,35,34,15,24,36);(2,4)(5,8)(6,14,16,34)(7,20)(9,23)(10,25)(11,27)(12,26)(13,30,19,15)(17,24,18,31)(21,29)(22,28)(32,36,33,35);(5,9,11,28,12,29,10,20)(6,15,18,36,16,30,17,35)(7,8,23,27,22,26,21,25)(13,31,33,34,19,24,32,14)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257017, #72257167, #72257214, …
maximal normal subgroup of	#72257017, #72257167, #72257214, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262072);