C₄×#3428 #7625215

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₄×#3428

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator	C₂	C₂⁴×C₄²
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

13438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2,3,4);(5,6,7,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(11,18)(14,17)(19,24)(21,23);(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72280453
maximal normal subgroup of	#72280453

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532411);

C4×#3428 #7625215