#412773

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1026 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252625, #72252627, #72252628, …
maximal normal subgroup of	#72252625, #72252627, #72252628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319969);