#374116

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2555 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)(8,12,21,28)(9,13)(10,15)(11,18)(14,23)(17,30,32,36)(19,29)(20,26,34,27)(22,25)(24,33,35,31);(1,2)(3,4)(5,7,10,19)(6,11,15,25)(8,17,21,32)(9,22,23,18)(12,24,28,35)(13,16,14,29)(20,33,34,31)(26,30,27,36);(2,4)(5,8)(6,12,15,28)(7,17)(9,20)(10,21)(11,24,25,35)(13,26,14,27)(16,30,29,36)(18,31)(19,32)(22,33)(23,34);(6,13)(9,23)(11,16)(12,26)(14,15)(18,22)(20,34)(24,30)(25,29)(27,28)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237697, #72256257, #72256268, …
maximal normal subgroup of	#72237697, #72256257, #72256268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281312);