#377681

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 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)(7,16)(8,12,21,27)(9,13)(10,15)(11,19)(14,22)(17,29,30,36)(18,28)(20,25,33,26)(23,34,35,31)(24,32);(1,2)(3,4)(5,7,9,19,10,18,22,32)(6,11,14,16,15,24,13,28)(8,17,20,31,21,30,33,34)(12,23,26,29,27,35,25,36);(2,4)(5,8,9,20,10,21,22,33)(6,12,14,26,15,27,13,25)(7,17,19,31,18,30,32,34)(11,23,16,29,24,35,28,36);(6,13,15,14)(7,18)(11,16,24,28)(12,25,27,26)(17,30)(19,32)(23,29,35,36)(31,34)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=fh,e^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,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250732, #72252977, #72256865
maximal normal subgroup of	#72250732, #72252977, #72256865

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284877);