#370002

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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)(7,12)(8,20)(9,25)(10,14)(11,15)(13,23)(16,33)(17,19)(18,29)(21,27)(22,32)(24,31)(26,28)(30,35)(34,36);(1,3,4,2)(5,7,10,18,11,19,9,26)(6,12,14,29,15,17,25,28)(8,21,24,35,22,36,23,16)(13,30,20,34,31,33,32,27);(5,8,10,23,11,22,9,24)(6,13,14,32,15,31,25,20)(7,16,18,36,19,35,26,21)(12,27,29,33,17,34,28,30);(5,6)(7,17)(8,13)(9,25)(10,14)(11,15)(12,19)(16,34)(18,28)(20,24)(21,33)(22,31)(23,32)(26,29)(27,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307083, #72307310, #72307320
maximal normal subgroup of	#72307083, #72307310, #72307320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277198);