#369552

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276748);