#387438

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1148 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)(7,17)(8,23)(9,26)(10,27)(11,15)(12,16)(13,20)(14,25)(18,33)(19,28)(21,29)(22,32)(24,31)(30,35)(34,36);(5,7,12,22)(6,13,16,29)(8,19,25,34)(9,18,24,35)(10,21,11,20)(14,30,23,33)(15,32,27,17)(26,36,31,28);(5,8,10,24,12,25,11,9)(6,14,15,31,16,23,27,26)(7,18,20,34,22,35,21,19)(13,28,17,30,29,36,32,33);(5,9,11,25,12,24,10,8)(6,14,15,31,16,23,27,26)(7,19,21,35,22,34,20,18)(13,28,17,30,29,36,32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306448, #72306450, #72306464
maximal normal subgroup of	#72306448, #72306450, #72306464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294634);