#375757

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312207, #72312221, #72312227
maximal normal subgroup of	#72312207, #72312221, #72312227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282953);