#494924

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

774 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)(6,12,15,16)(7,18,21,20)(8,13)(9,24)(10,23)(14,30)(17,28,26,27)(19,31)(22,29,34,35)(33,36);(5,6,13,28,23,26,8,16,11,15,24,27,10,17,9,12)(7,19,35,32,34,14,18,33,21,30,29,25,22,31,20,36);(2,4)(5,7,8,20,10,22,13,29,11,21,9,18,23,34,24,35)(6,14,12,25,17,19,27,33,15,31,16,32,26,30,28,36);(6,15)(12,16)(14,31)(17,26)(19,30)(25,32)(27,28)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249916, #72249923, #72249962, …
maximal normal subgroup of	#72249916, #72249923, #72249962, …
maximal quotient of	#72319078, #72319906

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402120);