#6497700

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470867
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3184	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282616, #72282676, #72282874
maximal normal subgroup of	#72282616, #72282676, #72282874

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404896);