#364595

order	512
nilpotency class	3
isoclinism	isoclinic class of #23882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2466 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,e^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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286523, #72287003, #72287104, …
maximal normal subgroup of	#72286523, #72287003, #72287104, …
maximal quotient of	#72314896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271791);