#6371561

order	512
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3487	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

6228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,d^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,g^b=gi,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306371, #72306520, #72318131
maximal normal subgroup of	#72306371, #72306520, #72318131

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278757);