#421699

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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

848 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)(5,6,10,16,11,17,12,29)(7,18,21,32,23,35,22,13)(8,24,9,15,26,14,27,33)(19,31,20,34,28,36,25,30);(5,7,12,21,11,23,10,22)(6,13,29,18,17,32,16,35)(8,25,27,19,26,20,9,28)(14,31,15,34,24,36,33,30);(5,8,10,9,11,26,12,27)(6,14,16,33,17,24,29,15)(7,19,21,20,23,28,22,25)(13,30,18,31,32,34,35,36);(2,4)(5,9,10,26,11,27,12,8)(6,15,16,14,17,33,29,24)(7,20,21,28,23,25,22,19)(13,31,18,34,32,36,35,30)

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,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328895);