#479910

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10,12,17,13,18,24,33,14,16,25,30,26,31,39,40)(11,19,21,35,22,28,37,32,20,34,23,38,36,15,27,29);(1,2,3,5,4,6,7,8)(9,11)(10,15,16,28)(12,23)(13,22)(14,20)(17,32,30,29)(18,19,31,34)(21,25)(24,27)(26,36)(33,38,40,35)(37,39);(11,20)(15,28)(19,34)(21,23)(22,36)(27,37)(29,32)(35,38);(10,16)(15,28)(17,30)(18,31)(19,34)(29,32)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,g^2=i,b^a=bg,d^a=di,f^a=fi,c^b=ci,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224814, #72224819, #72224820, …
maximal normal subgroup of	#72224814, #72224819, #72224820, …
maximal quotient of	#72227012, #72227018

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387106);