#227968

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=dei,e^b=eh,d^c=di,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135164);