#366453

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249926, #72249938, #72249969, …
maximal normal subgroup of	#72249926, #72249938, #72249969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273649);