#421698

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

884 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)(5,6,9,15,10,16,11,27)(7,17,20,30,22,35,21,12)(8,14,24,34,25,13,26,32)(18,31,36,29,19,33,23,28);(5,7,11,20,10,22,9,21)(6,12,27,17,16,30,15,35)(8,23,26,18,25,36,24,19)(13,31,34,29,14,33,32,28);(5,8,9,24,10,25,11,26)(6,13,15,32,16,14,27,34)(7,18,20,36,22,19,21,23)(12,28,17,31,30,29,35,33);(2,4)(5,8,9,24,10,25,11,26)(6,14,15,34,16,13,27,32)(7,19,20,23,22,18,21,36)(12,29,17,33,30,28,35,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252906, #72252972, #72252980
maximal normal subgroup of	#72252906, #72252972, #72252980

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328894);