#392141

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×H(7,𝔽₂)	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

16013 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:

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

Polycyclic presentation:

2^9|a^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72341955, #72341968, #72342003, …
maximal subgroup of	#72249722, #72268005, #72271658, …
maximal normal subgroup of	#72249722, #72268005, #72280569, …
maximal quotient of	#72252067, #72281532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299337);