#147024

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1799
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,f^b=fh,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240708, #72240802, #72240816, …
maximal normal subgroup of	#72240708, #72240802, #72240816, …
maximal quotient of	#72240712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54220);