#394265

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2638 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286018, #72286186, #72286194, …
maximal normal subgroup of	#72286018, #72286186, #72286194, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301461);