#373700

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2220 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307225, #72307343, #72307350
maximal normal subgroup of	#72307225, #72307343, #72307350

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280896);