#238734

order	512
nilpotency class	3
isoclinism	isoclinic class of #233984
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35236	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213359
maximal normal subgroup of	#72213359

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145930);